HW4_2006_Fall_solution

HW4_2006_Fall_solution - Solution to Homework 4 Acoustics I...

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Solution to Homework 4 Acoustics I 2006, Fall Qt. Problem 1 We want a design analysis for a reactive type muffler to be used for a small automobile. The first-cut model is shown below. Using the 4 pole approach : Figure 1. Simple Expansion Chamber Type Reactive Muffler d=5 cm D=15 cm d=5 cm 30 cm 10 cm 200 o C, 1 atm A B A Engine side For the frequency range between 20 – 3,000 Hz, (1) Calculate and plot the TL of the muffler proper (2) Calculate and plot the approximate insertion loss 10 20log A B Q IL Q = . (3) Discuss the results. (Solution) At the temperature is at 473 o K, the speed of sound becomes: 331 473/ 273 436 / cm s = . 3 273 1.293 0.745 / 473 Kg m ρ = , 324 o c = . (1) TL: The expansion chamber is connected to an anechoic termination as seen in Figure 2. Figure 2. TL setup d=5 cm D=15 cm d=5 cm 30 cm 10 cm 200 o C, 1 atm A B Engine side
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The four pole equation becomes: 11 1 1 1 2 1 sin cos 1 sin cos iB A B iB A jS kL kL Q Q AB c Q c jc k L CD P P kL S S ρ ⎡⎤ ⎢⎥ ⎧⎫ ⎪⎪ == ⎨⎬ ⎣⎦ ⎩⎭ % % % % % (1) The pressure reflection coefficient is calculated as: () 2 2 2 2 1 1 1 1 1 ir B an a n B a n B an an S PP c Q c Z S c Z Q S PP CD Z Rc RS C D Z R φ = = + ⎩⎭ ⎩⎭ −= + += + + ++ = + % The TL is calculated as: 2 10 10log (1 ) TL R =− (2) (2) Approximate Insertion Loss 2 2 12 2 1 2 2 10 1 1 where, ( ) 0.6 4 20log A B TT B rad B A o rad A r a d B A B Q Q AB A B A B Q Z CDCD P P c Zk a j k a S Q Z Q Q IL Q ⎡⎤⎡ ⎨ ⎬ ⎢⎥⎢ ⎣⎦⎣ ⎛⎞ =+ ⎜⎟ ⎝⎠ = % % % (3) Figure 3 below plots IL which is compared with the TL. ILs obtained by modeling the tail pipe as a short pipe (lumped parameter) and 1 D pipe are compared with each other. It is seen that one deviates from the other as the frequency increases.
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0 500 1000 1500 2000 2500 3000 -40 -20 0 20 40 60 frequency (Hz) 20 log10 (QA/QB) tail pipe modeled as a 1-D pipe
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HW4_2006_Fall_solution - Solution to Homework 4 Acoustics I...

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